Enveloping manifolds

AbstractWe study the problem of embedding compact subsets of Rn into C1 submanifolds of minimal dimension. In [Ott and Yorke, SIAM J. Appl. Dynamical Systems 2 (2003) 297], we define a generalized tangent space TxA suitable for a general compact subset A of Rn and we prove that A may be locally embedded into a C1 manifold of dimension dim(TxA). This result leads naturally to the global conjecture that for a compact subset A of Rn, there exists a C1 manifold M such that M⊃A and dimM=maxx∈Adim(TxA). We prove that this conjecture is false in general, but true if dim(TxA) is constant on A. Applications of these ideas to dimension theory, embedding theory, and dynamical systems are discussed